- FindStat

Identifier

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000071: Posets ⟶ ℤ

Values

[1] => [[1],[]] => ([],1) => 1

[1,1] => [[1,1],[]] => ([(0,1)],2) => 1

[2] => [[2],[]] => ([(0,1)],2) => 1

[1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 1

[1,2] => [[2,1],[]] => ([(0,1),(0,2)],3) => 2

[2,1] => [[2,2],[1]] => ([(0,2),(1,2)],3) => 2

[3] => [[3],[]] => ([(0,2),(2,1)],3) => 1

[1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 1

[1,1,2] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => 2

[1,2,1] => [[2,2,1],[1]] => ([(0,3),(1,2),(1,3)],4) => 3

[1,3] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => 2

[2,1,1] => [[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4) => 2

[2,2] => [[3,2],[1]] => ([(0,3),(1,2),(1,3)],4) => 3

[3,1] => [[3,3],[2]] => ([(0,3),(1,2),(2,3)],4) => 2

[4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 1

[1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[1,1,1,2] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 2

[1,1,2,1] => [[2,2,1,1],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 3

[1,1,3] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => 2

[1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 3

[1,2,2] => [[3,2,1],[1]] => ([(0,3),(0,4),(1,2),(1,4)],5) => 4

[1,3,1] => [[3,3,1],[2]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 3

[1,4] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 2

[2,1,1,1] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2

[2,1,2] => [[3,2,2],[1,1]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 3

[2,2,1] => [[3,3,2],[2,1]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 4

[2,3] => [[4,2],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 3

[3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[3,2] => [[4,3],[2]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 3

[4,1] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2

[5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

[1,1,1,1,2] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 2

[1,1,1,2,1] => [[2,2,1,1,1],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6) => 3

[1,1,1,3] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 2

[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6) => 3

[1,1,2,2] => [[3,2,1,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 4

[1,1,3,1] => [[3,3,1,1],[2]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6) => 3

[1,1,4] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 2

[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => 3

[1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 4

[1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 5

[1,2,3] => [[4,2,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 4

[1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 3

[1,3,2] => [[4,3,1],[2]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 4

[1,4,1] => [[4,4,1],[3]] => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6) => 3

[1,5] => [[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 2

[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 2

[2,1,1,2] => [[3,2,2,2],[1,1,1]] => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6) => 3

[2,1,2,1] => [[3,3,2,2],[2,1,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4

[2,1,3] => [[4,2,2],[1,1]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6) => 3

[2,2,1,1] => [[3,3,3,2],[2,2,1]] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6) => 4

[2,2,2] => [[4,3,2],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 5

[2,3,1] => [[4,4,2],[3,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 4

[2,4] => [[5,2],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6) => 3

[3,1,1,1] => [[3,3,3,3],[2,2,2]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 2

[3,1,2] => [[4,3,3],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 3

[3,2,1] => [[4,4,3],[3,2]] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6) => 4

[3,3] => [[5,3],[2]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6) => 3

[4,1,1] => [[4,4,4],[3,3]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 2

[4,2] => [[5,4],[3]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => 3

[5,1] => [[5,5],[4]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 2

[6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1

[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1

[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => 2

[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7) => 3

[1,1,1,1,3] => [[3,1,1,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 2

[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7) => 3

[1,1,1,2,2] => [[3,2,1,1,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7) => 4

[1,1,1,3,1] => [[3,3,1,1,1],[2]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7) => 3

[1,1,1,4] => [[4,1,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7) => 2

[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7) => 3

[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7) => 4

[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7) => 5

[1,1,2,3] => [[4,2,1,1],[1]] => ([(0,5),(0,6),(1,4),(1,6),(4,2),(5,3)],7) => 4

[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7) => 3

[1,1,3,2] => [[4,3,1,1],[2]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7) => 4

[1,1,4,1] => [[4,4,1,1],[3]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7) => 3

[1,1,5] => [[5,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 2

[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7) => 3

[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7) => 4

[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7) => 5

[1,2,1,3] => [[4,2,2,1],[1,1]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7) => 4

[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7) => 5

[1,2,2,2] => [[4,3,2,1],[2,1]] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5)],7) => 6

[1,2,3,1] => [[4,4,2,1],[3,1]] => ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7) => 5

[1,2,4] => [[5,2,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7) => 4

[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => ([(0,5),(1,2),(1,4),(3,6),(4,6),(5,3)],7) => 3

[1,3,1,2] => [[4,3,3,1],[2,2]] => ([(0,3),(0,5),(1,2),(1,4),(4,6),(5,6)],7) => 4

[1,3,2,1] => [[4,4,3,1],[3,2]] => ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7) => 5

[1,3,3] => [[5,3,1],[2]] => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7) => 4

[1,4,1,1] => [[4,4,4,1],[3,3]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7) => 3

[1,4,2] => [[5,4,1],[3]] => ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7) => 4

[1,5,1] => [[5,5,1],[4]] => ([(0,6),(1,2),(1,5),(3,6),(4,3),(5,4)],7) => 3

[1,6] => [[6,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => 2

[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => 2

[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => ([(0,6),(1,2),(1,5),(3,6),(4,3),(5,4)],7) => 3

[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7) => 4

[2,1,1,3] => [[4,2,2,2],[1,1,1]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7) => 3

[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7) => 4

[2,1,2,2] => [[4,3,2,2],[2,1,1]] => ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7) => 5

>>> Load all 154 entries. <<<

[2,1,3,1] => [[4,4,2,2],[3,1,1]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 4

[2,1,4] => [[5,2,2],[1,1]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7) => 3

[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7) => 4

[2,2,1,2] => [[4,3,3,2],[2,2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7) => 5

[2,2,2,1] => [[4,4,3,2],[3,2,1]] => ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7) => 6

[2,2,3] => [[5,3,2],[2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7) => 5

[2,3,1,1] => [[4,4,4,2],[3,3,1]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7) => 4

[2,3,2] => [[5,4,2],[3,1]] => ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7) => 5

[2,4,1] => [[5,5,2],[4,1]] => ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7) => 4

[2,5] => [[6,2],[1]] => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7) => 3

[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => 2

[3,1,1,2] => [[4,3,3,3],[2,2,2]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7) => 3

[3,1,2,1] => [[4,4,3,3],[3,2,2]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7) => 4

[3,1,3] => [[5,3,3],[2,2]] => ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7) => 3

[3,2,1,1] => [[4,4,4,3],[3,3,2]] => ([(0,5),(0,6),(1,4),(2,3),(3,5),(4,6)],7) => 4

[3,2,2] => [[5,4,3],[3,2]] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7) => 5

[3,3,1] => [[5,5,3],[4,2]] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7) => 4

[3,4] => [[6,3],[2]] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7) => 3

[4,1,1,1] => [[4,4,4,4],[3,3,3]] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7) => 2

[4,1,2] => [[5,4,4],[3,3]] => ([(0,5),(1,2),(1,4),(3,6),(4,6),(5,3)],7) => 3

[4,2,1] => [[5,5,4],[4,3]] => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7) => 4

[4,3] => [[6,4],[3]] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7) => 3

[5,1,1] => [[5,5,5],[4,4]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => 2

[5,2] => [[6,5],[4]] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7) => 3

[6,1] => [[6,6],[5]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => 2

[7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1

[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 1

[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8) => 2

[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8) => 2

[1,1,1,1,4] => [[4,1,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8) => 2

[1,1,1,5] => [[5,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8) => 2

[1,1,6] => [[6,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8) => 2

[1,7] => [[7,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8) => 2

[8] => [[8],[]] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 1

[1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => 1

[1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) => 2

[1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9) => 2

[1,1,1,1,1,4] => [[4,1,1,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9) => 2

[1,1,1,1,5] => [[5,1,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9) => 2

[1,1,1,6] => [[6,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9) => 2

[1,1,7] => [[7,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9) => 2

[1,8] => [[8,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) => 2

[9] => [[9],[]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => 1

[1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1],[]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

[1,1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1,1],[]] => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 2

[1,1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,1)],10) => 2

[1,1,1,1,1,1,4] => [[4,1,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,4),(4,6),(5,3),(6,2),(7,1),(8,7),(9,5)],10) => 2

[1,1,1,1,1,5] => [[5,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10) => 2

[1,1,1,1,6] => [[6,1,1,1,1],[]] => ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10) => 2

[1,1,1,7] => [[7,1,1,1],[]] => ([(0,8),(0,9),(3,4),(4,6),(5,3),(6,2),(7,1),(8,7),(9,5)],10) => 2

[1,1,8] => [[8,1,1],[]] => ([(0,8),(0,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,1)],10) => 2

[1,9] => [[9,1],[]] => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 2

[10] => [[10],[]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 2\ q + 2\ q^{2}$

$F_{4} = 2\ q + 4\ q^{2} + 2\ q^{3}$

$F_{5} = 2\ q + 6\ q^{2} + 6\ q^{3} + 2\ q^{4}$

$F_{6} = 2\ q + 8\ q^{2} + 12\ q^{3} + 8\ q^{4} + 2\ q^{5}$

$F_{7} = 2\ q + 10\ q^{2} + 20\ q^{3} + 20\ q^{4} + 10\ q^{5} + 2\ q^{6}$

Description

The number of maximal chains in a poset.

Map

to ribbon

Description

The ribbon shape corresponding to an integer composition.
For an integer composition

$(a_1, \dots, a_n)$ , this is the ribbon shape whose

$i$ th row from the bottom has

$a_i$ cells.

Map

cell poset

Description

The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell

$d$ is greater than a cell

$c$ if the entry in

$d$ must be larger than the entry of

$c$ in any standard Young tableau on the skew partition.